Once you introduce that, you have SIA. If you can talk about having a 50% chance of being the observer who is shown heads, wouldn’t you also have a specific chance of being the observer who wakes up in a room running that experiment?
Looking into it more, it seems SSA is the one I agree with. I just always assume that there are more people in her reference class that aren’t in the experiment, so I get a different answer than what Wikipedia gave. No wonder I “got them confused” at first.
How exactly do you get 50% in this thought experiment?
Okay. Read this line carefully. I’m taking it straight from Wikipedia and you linked to the Wikipedia articles, so you must have read it. “All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class.”
In this case, conditioned on the two coinflips being TH, the observers in the reference class are all the observers in the TH world, but after waking up in the experiment, we know that only two possible observers remain: Beauty waking up on Monday, and Beauty waking up on Tuesday. We have no information to select one of these over the other and so each is 50% likely. Therefore the probability of being shown heads, given that the two coinflips are TH, is 50% by the SSA.
The difference between SSA and SIA is that in SSA we first randomly pick one of the possible worlds, and then pick one of the possible observers in that world. In SIA, we randomly pick one of the possible observers in all worlds.
Edit: to emphasize—it is irrelevant how many other observers there are besides Sleeping Beauty. Once Beauty wakes up and looks around, she knows that she is Sleeping Beauty as opposed to, say, the person running the experiment. However, she has no additional information on which instance of Sleeping Beauty she is, which is what the thought experiment is all about.
We could postulate some additional number of observers that wake up in the same situation for completely different reasons—say, someone else is running a simulation of lots of people waking up. In that case, the probabilities of 1⁄3 and 1⁄2 are conditional probabilities—conditional on Sleeping Beauty actually being part of this experiment. In the original formulation of the problem, we do not postulate these additional observers waking up—because their existence is independent of the coin flips, including them or not does not differentiate between the SIA and the SSA, so we either don’t think about them or we deal with the conditional probabilities.
What I’m arguing against is apparently neither SIA or SSA. I made a mistake. Are we arguing about what I originally intended to, or about my statement that they both predict 1/3?
My intent was to argue the, if you only update on the existence of an observer, rather than anything about how unlikely it is to be them, the probability will work out the same.
If you would like to discuss why SIA and SSA give the same result:
For simplicity, we’ll assume 1 trillion observation days outside the experiment.
SIA: Sleeping beauty wakes up in this experiment. There are 2 trillion 3 possible observers, three of which wake up here. Of them, one woke up in a universe with heads, and the other in a universe with tails. The probability of being the one with heads is 1⁄3.
SSA: Sleeping beauty has an even prior, so the odds ratio of heads to tails is 1:1. She then wakes up in this experiment. If the coin landed on heads, there’s a 1⁄1 trillion 1 chance of this. If the coin landed on tails, there’s a 2⁄1 trillion 2 chance of this. This is an odds ratio of 500,000,000,001:1,000,000,000,001 for heads. Multiplying this by 1:1 yields 500,000,000,001:1,000,000,000,001. The total probability of heads is 1⁄3 + 2*10^-13.
Your problem seems to be updating on the fact that she’s in the experiment without taking into account that this is about twice as likely if the coin landed on heads.
...huh. You have a point. I’ll have to think about this for a bit, but it seems right, and if this is what you’ve been trying to get at this whole time I think everyone may have misunderstood you.
Once you introduce that, you have SIA. If you can talk about having a 50% chance of being the observer who is shown heads, wouldn’t you also have a specific chance of being the observer who wakes up in a room running that experiment?
Looking into it more, it seems SSA is the one I agree with. I just always assume that there are more people in her reference class that aren’t in the experiment, so I get a different answer than what Wikipedia gave. No wonder I “got them confused” at first.
How exactly do you get 50% in this thought experiment?
Okay. Read this line carefully. I’m taking it straight from Wikipedia and you linked to the Wikipedia articles, so you must have read it. “All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class.”
In this case, conditioned on the two coinflips being TH, the observers in the reference class are all the observers in the TH world, but after waking up in the experiment, we know that only two possible observers remain: Beauty waking up on Monday, and Beauty waking up on Tuesday. We have no information to select one of these over the other and so each is 50% likely. Therefore the probability of being shown heads, given that the two coinflips are TH, is 50% by the SSA.
The difference between SSA and SIA is that in SSA we first randomly pick one of the possible worlds, and then pick one of the possible observers in that world. In SIA, we randomly pick one of the possible observers in all worlds.
Edit: to emphasize—it is irrelevant how many other observers there are besides Sleeping Beauty. Once Beauty wakes up and looks around, she knows that she is Sleeping Beauty as opposed to, say, the person running the experiment. However, she has no additional information on which instance of Sleeping Beauty she is, which is what the thought experiment is all about.
We could postulate some additional number of observers that wake up in the same situation for completely different reasons—say, someone else is running a simulation of lots of people waking up. In that case, the probabilities of 1⁄3 and 1⁄2 are conditional probabilities—conditional on Sleeping Beauty actually being part of this experiment. In the original formulation of the problem, we do not postulate these additional observers waking up—because their existence is independent of the coin flips, including them or not does not differentiate between the SIA and the SSA, so we either don’t think about them or we deal with the conditional probabilities.
What I’m arguing against is apparently neither SIA or SSA. I made a mistake. Are we arguing about what I originally intended to, or about my statement that they both predict 1/3?
My intent was to argue the, if you only update on the existence of an observer, rather than anything about how unlikely it is to be them, the probability will work out the same.
If you would like to discuss why SIA and SSA give the same result:
For simplicity, we’ll assume 1 trillion observation days outside the experiment.
SIA: Sleeping beauty wakes up in this experiment. There are 2 trillion 3 possible observers, three of which wake up here. Of them, one woke up in a universe with heads, and the other in a universe with tails. The probability of being the one with heads is 1⁄3.
SSA: Sleeping beauty has an even prior, so the odds ratio of heads to tails is 1:1. She then wakes up in this experiment. If the coin landed on heads, there’s a 1⁄1 trillion 1 chance of this. If the coin landed on tails, there’s a 2⁄1 trillion 2 chance of this. This is an odds ratio of 500,000,000,001:1,000,000,000,001 for heads. Multiplying this by 1:1 yields 500,000,000,001:1,000,000,000,001. The total probability of heads is 1⁄3 + 2*10^-13.
Your problem seems to be updating on the fact that she’s in the experiment without taking into account that this is about twice as likely if the coin landed on heads.
...huh. You have a point. I’ll have to think about this for a bit, but it seems right, and if this is what you’ve been trying to get at this whole time I think everyone may have misunderstood you.